A conditional limit theorem for random walks under extreme deviation

Michel Broniatowski (1) , Zhansheng Cao (1)

1 LSTA - Laboratoire de Statistique Théorique et Appliquée

This paper explores a conditional Gibbs theorem for a random walkinduced by i.i.d. (X₁,..,X_{n}) conditioned on an extreme deviation of its sum (S₁ⁿ=na_{n}) or (S₁ⁿ>na_{n}) where a_{n}→∞. It is proved that when the summands have light tails with some additional regulatity property, then the asymptotic conditional distribution of X₁ can be approximated in variation norm by the tilted distribution at point a_{n} , extending therefore the classical LDP case.

Mots clés: Extreme deviation; Large deviation; Gibbs conditional theorem; random walk

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Probabilités [math.PR]

Michel Broniatowski, Zhansheng Cao. A conditional limit theorem for random walks under extreme deviation. 2012. ⟨hal-00713053⟩